Essentially Nonperturbative Vacuum Polarization Effects in a Two-Dimensional Dirac-Coulomb System for Z > Zcr: Vacuum Polarization Effects

For a planar Dirac-Coulomb system with a supercritical extended axially symmetric Coulomb source with a charge Z > Zcr,1and radius R0, we consider essentially nonperturbative vacuum polarization effects in the overcritical region. Using results obtained in our preceding paper for the induced charge density ρVP(r→)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rho _{{\rm{VP}}}}(\vec r)$\end{document}, we thoroughly consider the calculation of the vacuum energy εVPbased on the renormalization, the convergence of the partial expansion for ρVP(r→)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rho _{{\rm{VP}}}}(\vec r)$\end{document}, and the behavior of the integral induced charge QVPin the overcritical region. In particular, we show that the renormalization using the fermionic loop with two external lines turns out to be a universal technique, which eliminates the divergence of the theory in the purely perturbative and essentially nonperturbative modes for ρVP(r→)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rho _{{\rm{VP}}}}(\vec r)$\end{document} and εVP. The most significant result is that for Z ≫ Zcr,1in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge Z reaching large negative values; its behavior is estimated from below (in absolute value) as ∼ −|ηeffZ3|/R0. We also study the dependence of the effect of the decrease in εVPon the cutoff of the Coulomb asymptotics of the external field at different scales R1 gt; R0and R1 ≫ R0.